\newproblem{lay:7_1_13}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 7.1.13}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Orthogonalize the matrix $A=\begin{pmatrix} 3 & 1\\ 1& 3 \end{pmatrix}$ giving a matrix $P$ and a diagonal matrix $D$.
}{
   % Solution
	Let's find first the eigenvalues of $A$
	\begin{center}
		$|A-\lambda I|=0$ \\
		$\left|\begin{array}{cc}3 -\lambda& 1 \\ 1 & 3-\lambda \end{array}\right|=(3-\lambda)^2-1=\lambda^2-6\lambda+8=(\lambda-2)(\lambda-4)=0$
	\end{center}
	Let's find now the eigenvalues associated to each eigenspace\\
	\underline{Eigenspace $\lambda=2$}\\
	Let's solve the vector problem
	\begin{center}
		$(A-2I)\mathbf{v}=\mathbf{0}$ \\
		$\begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}\mathbf{v}=\mathbf{0}$
	\end{center}
	whose solution are all vectors of the form $\mathbf{v}=(v_1,-v_1)$. In particular $\mathbf{v}_1=(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$ is  unit vector
	of this subspace.\\
	\underline{Eigenspace $\lambda=4$}\\
	\begin{center}
		$(A-4I)\mathbf{v}=\mathbf{0}$ \\
		$\begin{pmatrix}-1 & 1 \\ 1 & -1\end{pmatrix}\mathbf{v}=\mathbf{0}$
	\end{center}
	whose solution are all vectors of the form $\mathbf{v}=(v_1,v_1)$. In particular $\mathbf{v}_2=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$ is  unit vector
	of this subspace.\\
	The eigendecomposition of matrix $A$ is, therefore,
	\begin{center}
		$A=PDP^{-1}$ \\
		$\begin{pmatrix} 3 & 1\\ 1& 3 \end{pmatrix}=\begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}} \end{pmatrix}
		      \begin{pmatrix} 2 & 0\\ 0& 4 \end{pmatrix}
					\begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}} \end{pmatrix}$
	\end{center}
	Note that we can find an orthogonal matrix for $P$ (and consequently $P^{-1}=P^T$) because $A$ is a symmetric.
}
\useproblem{lay:7_1_13}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}

